High $\mathrm{\ell }$-torsion rank for class groups over function fields

We prove that in the function field setting, $\mathrm{\ell }$-torsion in the class groups of quadratic fields can be arbitrarily large. In fact, we explicitly produce a family whose $\mathrm{\ell }$-rank growth matches the growth in the setting of genus theory, which might be best possible. We do this by specifically focusing on the Artin-Schreir curves $y^2=x^q-x$.